It's been a little while since my differential equations class and I'm not sure where the k3 comes from. Is it because the equation is nonhomogeneous and if that's the case will all second order nonhomogeneous equations that equal a constant have similar general equations?

In my opinion, that is really bad notation! The fact that the same symbol is used for [itex]k_1, k_2, k_3[/itex] implies that they are to be treated the same. In fact, k1 and k2 are arbitrary constants while k3 must be 8.